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Full text of "Treatise On Analysis Vol-Ii"

418        XV   NORMED ALGEBRAS AND SPECTRAL THEORY

En . Then there exists a unique normal operator N on E such that En c dom(TV)
and Nn = N\En, for all n. The set F = dom(TV) is the set of all x = ]T xn

n

(where xn e Enfor all n) such that

and we have

N-x=£Nn-xn       and       N*-x^N*-xn.

n                                                                               n

We shall begin by proving the first assertion of (i). We have seen in the
course of the proof of (15.12.6) that the graph of the restriction of N to
dom(N*N) is dense in T(N); hence, for each xedom(N), there exists a
sequence (yj in dom(N*N) such that lim yn = x and lim N • yn = TV* x.

«-» 00                                   7J-* 00

But for each z e dom(N*N) we have

\\N-z\\2 = (z\ N*N - z) = (z\NN* - z) = ||JV* • z||2,

because N*N = NN*. Applying this result with z = yn - ym , it follows
that the sequence (N* • .yj is a Cauchy sequence and therefore converges
in E. Since N* is closed (15.12.4), we conclude that xedom(N*) and
N*-x = lim TV* • yn, whence ||JV • #|| = ||7V* • jc||. We have therefore proved

n-» oo

that dom(N) c dom(^V*). Since TV** = N9 the operator N* is normal and
dom(^*) c dom(A^).

Now consider (ii). We shall first show that, if N satisfies the conditions
of (ii), then N*(En) c En . For all m^ny all yn e En and all xm e Em , we
have (xm \ N* - yn) = (N - xm\yn) = 0, because N(EJ c Em . Hence N* • yn is
orthogonal to all the Em (m ^ n\ hence must lie in En . Next we shall show that,
if x E dom(JV), then

PEn AT • x = JVPEn • x = Nn - (PEn - x) .
For all yn e En , we have

(PEnN 'X\yn)=(N-x\PEn' yn) =(N-x\yJ=(x\N*- yn\
and since TV* • yn = PEn JV* • jn from above,

(PEnN -x\yn)-(PEn -xlN*- yn) = (NPEn 'X\yn)^(Nn - (PEn - x)\yj;

since this holds for all yn e Ew, it follows that the two elements PEnN • x and
•W» * (^En " x) °f E are equal.

We shall now prove the following lemma, which generalizes (15.10.8.1):f N is a not necessarily bounded normal operator, then dom(TV) =